We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric \(Q_{2} \cong \mathbb {S}^{2}\times \mathbb {S}^{2}\) , formulated via a flat family of connections \(\{\nabla ^\lambda \}_{\lambda \in \mathbb {S}^{1}}\) on a trivial bundle. We prove that minimality is equivalent to the flatness of \(\nabla ^\lambda \) for all \(\lambda \) , describe the associated isometric \(\mathbb {S}^{1}\) -family, and establish a precise correspondence with minimal surfaces in \(\mathbb {S}^{3}\) through their Gauss maps. Our framework unifies and streamlines earlier constructions (e.g., Castro–Urbano) and yields explicit families including \(\mathbb {R}\) -equivariant, radially symmetric, and trinoid-type examples.
